Methods and apparatus for investigating earth formations

ABSTRACT

In accordance with illustrative embodiments of the present invention, an electrode array is utilized for investigating earth formations surrounding a mud filled borehole without undue influence from a conductive mud column. One manner of accomplishing this is to establish a potential distribution in the borehole as if the mud column were of the same resistivity as the formation. Alternatively, a zero potential gradient can be established at various regions in the mud column to effectively block current flow up or down the mud column.

OR 3;, 793 a--- united State Schuster 1 1 METHODS AND APPARATUS FOR INVESTIGATING EARTH FORMATIONS [75] Inventor: Nick A. Schuster, Darien, Conn.

[73] Assignee: Schlumberger Technology Corporation, New York, N.Y.

[22] Filed: Apr. 24, 1972 [21] Appl. No.: 247,199

Related US. Application Data [62] Division of Ser. No. 815265, April 7, 1969.

abandoned.

[52] U.S. Cl. 324/10 [51] Int. Cl G0lv 3/18 5 [58] Field of Search 324/1, 10

[56] References Cited UNITED STATES PATENTS 2,806.20] 9/1957 Schuster 324/10 X 2,872,637 2/1959 Blanchard 324/1 [451 Mar. 19, 1974 Primary ExaminerGerard R. Strecker Attorney, Agent, or Firm-Amster & Rothstein 5 7 ABSTRACT 3 Claims, 17 Drawing Figures 70 SUHFHCE NET WORK PATENTH] MAR I 9 I874 SHEE! 01 0F 10 PRIOR I? T PRIOR HBT 9 4 1 PAIENTEDMAR 19 I974 SHEET 03 0F 10 HON PATENIEDHAR 19 1974 SHEET 08 HF l0 ELECTRON C CKTS PATENTEDMAR w 9 m4 SHEET US [If 10 PATENTH] UAR I 9 I974 saw 10 or 10 gay.

7'0 SURF/7C5 T I EFT/0 CIRCUIT P/YHSE SENSITIVE DETECTOR OSCILLH 70/? P/IHSE SENSITIVE DETEC 70R flMPL/F/[R MEASURE I I/EJQZ METHODS AND APPARATUS FOR INVESTIGATING EARTH FORMATIONS This is a division of application Ser. No. 815,265, filed Apr. 7, 1969, now abandoned.

This invention relates to methods and apparatus for investigating earth formations traversed by a borehole and more particularly to investigation of earth formations by so-called electrode type investigating devices.

It is common practice to investigate earth formations traversed by a borehole with electrode type investigating devices to obtain a log of the electrical resistivity or conductivity of the formations versus depth. One form of such an investigating device emits unfocused current from one electrode and records variations in the potential difference between two other potential measuring electrodes. These potential measuring electrodes can both be located near the current-emitting electrode, both located at a relatively great distance therefrom, or one measuring electrode located near the currentemitting electrode and one remotely located therefrom. The entire electrode array is continuously movable through a borehole so that variations in the measured potential difference can be recorded as a function of depth to provide indications of the resistivity or conductivity of the formations which formations include the rock matrix and formation fluids. Examples of such investigating devices are shown in US. Pat. No. 1,894,328 granted to C. Schlumberger on Jan. 17, 1933.

There are problems however which may arise when using the above-described electrode type exploring de vices under certain borehole conditions. One such problem occurs when the mud fluid fillingthe borehole, i.e., the mud column, is much more conductive than the adjacent formations. In this event, the current emitted from the principal current-emitting electrode will tend to flow primarily within the fluid in the borehole rather than within the formations. Thus, the potential measuring electrodes would, in this case, tend to be influenced more by the conductivity of the mud column than that of the formations, thus introducing errors into the measurements.

Another class of electrode type logging devices acts to focus a beam of so-called survey current into the formations for the purpose of, among other things, eliminating this borehole or mud column effect. ln this class of exploring devices, a pair of focusing electrodes is located on either side of a central survey current electrode. Current is emitted from all three electrodes, the current emitted from both focusing electrodes being in-phase with the survey current emitted from the survey electrode so as to focus the survey current into a thin current beam which extends deeply into the formations. Typical examples of such focused" electrode investigating devices are disclosed in US. Pat. Nos. 2,712,627 granted to H. G. Doll on July 5, 1955 and 3,031,612 granted to M. F. Easterling on Apr. 24,

1962. However, as is typical with these focused electrode exploring devices, the radial investigation (i.e., the investigation in a direction perpendicular to the borehole axis) tends to be relatively deep.

While both the focused and nonfocused exploring devices have heretofore provided good results, it would be desirable to have an additional electrode type exploring device which has a relatively shallow depth of investigation, like the nonfocused exploring devices, but yet is not undesirably influenced by a conductive mud column.

Also, it is often desirable to obtain a measure of the resistivity in formation zones at different radii with respect to the borehole axis and to do so with only one run through the borehole. One such tool for accomplishing this is the combination induction logging and focused electrode tool disclosed in US. Pat. No. 3,329,889 granted to D. R. Tanguy on July 4, 1967 which shows both coils and electrodes mounted on the same mandrel. While the radial depth of investigation of the focused electrode tool shown in the Tanguy patent is relatively shallow, it is known that an even shallower depth of investigation can be obtained with exploring devices which have most of the electrodes mounted on a borehole wall-engaging pad member. An example of this type of exploring device is disclosed in US. Pat. No. 3,379,965 granted to D. R. Tanguy et al on Apr. 23, 1968. However, it is difficult to utilize such a wall-engaging pad member simultaneously with an induction logging device because of the adverse effect that the metal portion of the pad and connecting arms have on the induction logging device.

Thus, it is an object of the present invention to provide an electrode type exploring device which is compatible with an induction logging device, yet has a radial depth of investigation comparable with a pad mounted exploring device. Along the same lines, it is another object of this invention to provide an electrode type exploring device that provides both shallow and deep radial investigation at the same time.

While the focused electrode type investigating devices discussed above are much less sensitive to a relatively conductive mud column than the nonfocused electrode type devices, if the mud column is substantially more conductive than the formation under investigation, the potential distribution produced by such a focused electrode device will tend to be altered from the desired distribution thus giving rise to a possibility for error. In this connection, it is another object of the present invention to provide a focused electrode device of the type described above which would not be unduly affected by a highly conductive mud column.

It is a further object of the present invention to provide new andimproved methods and apparatus for use with electrode type exploring devices which substantially minimizes the effect of a conductive mud column on the measurement of resistivity or conductivity.

In accordance with the present invention, a method for investigating earth formations penetrated by a borehole comprises moving a support means carrying a plurality of electrodes through a fluid-filled borehole. Survey and auxiliary currents are supplied to at least one of the electrodes for emission into a media surrounding the support means, the emitted currents returning to at least one current return electrode. The potential distribution in at least a portion of a borehole is measured and at least one of the survey or auxiliary currents is adjusted to maintain a desired potential distribution in said portion of the borehole. By so doing, the survey current will tend to be forced into a formation to enable a measure of a formation characteristic to be obtained.

In one form of the invention, one of the currents is adjusted to give a potential distribution through a borehole which decreases as an inverse function of distance.

This can be accomplished by controlling either the potential or potential gradient at two or more points in a borehole.

In another form of the invention, the potential distribution at only one location in a borehole is controlled. In a desirable manner, this control takes the form of maintaining the difference in potential between two points in a borehole at substantially zero volts so as to produce a borehole interval of zero potential gradient. In a representative embodiment of this zero gradient form of the invention, survey and auxiliary currents are emitted from a central current-emitting electrode. The auxiliary current is returned to a pair of current return electrodes located on each side of the central electrode and the survey current is returned to another electrode. The potential distribution at locations above and below the central and auxiliary current return electrodes is measured and utilized to adjust one of the survey or auxiliary currents until a desired potential distribution at these locations is established. By so doing, the auxiliary current will act to force the survey current out of the borehole and into the formations when the mud column is more conductive than the adjoining formations.

Apparatus for practicing the invention and the methods thereof is illustrated in several embodiments.

The methods and apparatus of the present invention can find utility in most any situation where a conductive medium surrounding or near the exploring device hampers the investigation of the formations adjoining the conductive medium. Thus, the present invention can find utility with exploring devices of the type where a plurality of electrodes are mounted on a borehole wall-engaging pad member so that the survey current will be prevented from shorting through a conductive mudcake. Additionally, the present invention can be used in combination with the prior art focused electrode systems to prevent a conductive mud column from having an adverse affect on the investigation operation.

In another form of the invention, reciprocal arrangements of the above-mentioned embodiments can be used.

For a better understanding of the present invention, together with other and further objects thereof, reference is had to the following description taken in connection with the accompanying drawings, the scope of the invention being pointed out in the appended claims.

Referring to the drawings:

FIG. 1 shows a prior art nonfocused type electrode array in a borehole along with a representation of equipotential lines and current flow which could be expected under certain conditions;

FIG. 2 shows the equipotential and current flow lines which could be expected under adverse conditions with the FIG. 1 electrode array;

FIG. 3 is a plot of potential versus distance along the borehole axis when using an electrode type investigating system for various borehole conditions;

FIG. 4 shows an electrode type exploring device in a borehole along with electrical circuitry for energizing the electrodes in accordance with the present invention;

FIG. 5 is a reproduction of one of the curves of FIG. 3 (corresponding to R Rf) for purposes of explaining certain features of the present invention;

FIG. 6 shows an electrode type exploring device in a borehole in accordance with another embodiment of the present invention;

FIG. 7 shows an exploring device in a borehole along with representations of the adjoining formations for the purpose of defining certain terms utilized in the explanation of the present invention;

FIG. 8 shows an electrode type exploring device in a borehole along with the accompanying electrical circuitry in accordance with another embodiment of the present invention;

FIG. 9 illustrates typical current flow patterns which could be expected from the investigating apparatus of FIG. 8;

FIG. 10 illustrates another embodiment of the present invention;

FIG. 1 1 illustrates the utilization of the present invention for the case where a portion of the electrodes are mounted on a wall-engaging pad member;

FIGS. 12 and 14 illustrate various electrode arrays that could be mounted on the pad member of FIG. 11 for investigating the earth formations adjoining the borehole;

FIG. 13 illustrates the current flow pattern which could be expected with the FIGS. 12 and 14 electrode arrays;

FIG. 15 illustrates a prior art focused electrode type investigating apparatus along with a typical plot of the equipotential lines and current flow pattern produced by such an electrode array;

FIG. 16 illustrates apparatus for improving the prior art investigating apparatus illustrated in FIG. 15 in accordance with the present invention; and

FIG. 17 illustrates how the apparatus of FIG. 8 can be modified to produce a reciprocal arrangement thereof.

Referring now to FIG. I, there is shown a prior art electrode array situated in a borehole 20 filled with a conductive drilling mud having a resistivity R, for investigating an adjacent formation 21 having a resistivity R,. The electrodes comprise a current-emitting electrode A and two potential measuring electrodes M and N. The current return electrode (not shown) is located at some considerable distance from the electrodes A, M and N. In this FIG. 1 situation, the mud resistivity R is considered to be substantially equal to the formation resistivity R, and thus the current flow (dotted lines) will be distributed evenly in all directions since the current return electrode is at electrical infinity.

With the known current distribution, the equipotential lines 22 and 23 passing through the measuring electrodes M and N can be drawn perpendicular to the cur rent flow lines. In three dimensional space, these equipotential lines 22 and 23 will have the form of spherical shells. By measuring the current emitted from the current emitting electrode A and the potential difference between the measure electrodes M and N, the resistivity of the shaded area of FIG. 1 can be determined. Thus, for example, assuming that the current emitted from electrode A is constant, the difference in the potentials measured by the electrodes M and N will vary in accordance with the resistivity of the shaded area of FIG. 1 through the application of Ohms Law.

Now referring to FIG. 2, there are shown equipotential lines 24, 25, and current flow lines (shown in dotted line form) when the mud resistivity R is substantially less than the formation resistivity R;. In this FIG. 2 case,

most of the current emitted from the electrode A will travel through the high conductivity borehole fluids and will not penetrate into the formation. Thus, knowing that the equipotential lines are perpendicular to the current flow lines, the equipotential lines 24 and 25 will take the form shown in FIG. 2. Thus, in this extreme case, the difference in potential between the measure electrodes M and N will be strongly effected by the value of the mud resistivity R In FIG. 3, there is shown a plot of voltage potential versus distance D along the borehole axis for various ratios of the mud resistivity R to the formation resistivity R,. If R,,, R the entire medium surrounding the electrodes is one electrically homogeneous volume. In this homogeneous case, the voltage potential V will decrease as an inverse function of the distance D from the current source along the borehole axis (i.e., V M V /D), where V is the potential at the current source and A is a constant characteristic of the current electrode A Thus, the R R, potential distribution curve of FIG. 3 corresponds to the spherical equipotential relationship shown in FIG. 1. As the mud resistivity becomes less than the formation resistivity, the potential versus distance curve tends to flatten out as illustrated in FIG. 3. This is as expected since the potential drop per unit distance through the borehole when the current flows through a highly conductive borehole or mud column will tend to be small.

From the above, with the electrode systems of FIGS. 1 and 2, it can be said that at any time the resistivity of the mud column is different from the resistivity of the adjoining formations, the equipotential lines will be dif ferent from the desired spherical distribution illustrated in FIG. 1. As is illustrated in FIG. 2, the resistivity of the mud column relative to that of the formation greatly affects the shape of the equipotential plot and thus greatly affects the determination of which portion of the formations and/or borehole is investigated. As was stated earlier, the prior art focused electrode exploring devices tend to minimize this mud column effeet, but at the same time, tend to produce relatively deep radial investigation.

It is a purpose of the present invention to provide novel methods and apparatus for investigating resistivity or conductivity characteristics of a formation regardless of the conductivity of the mud column. It is also desired with the present invention to provide an exploring device with relatively shallow radial investigation characteristics which is, at the same time, relatively unaffected by a conductive mud column. In one form of the present invention, this is accomplished by setting up a potential distribution through the borehole which will tend to approximate the spherical equipotential distribution illustrated in FIG. I regardless of the difference in resistivity between the mud column and formation. This is accomplished by enforcing a potential distribution through the borehole as if the resistivity R, of the mud column was substantially equal to the resistivity R, of the formation. As discussed in connection with FIG. 3, this potential distribution curve for the homogeneous case of R,,, R, is a potential curve which decreases as an inverse function of distance along the borehole axis. By enforcing this potential distribution along the borehole axis, the potential should decrease in the same fashion in the formation provided the formation itself is homogeneous.

Now turning to FIG. 4, there is shown apparatus for investigating the resistivity or conductivity of adjoining earth formations without being unduly affected by the relative conductivity of the mud. column. In the FIG. 4

apparatus, this is accomplished by controlling the potential at two points in the borehole in accordance with the homogeneous condition (R R,) curve of FIG. 3. More particularly, in FIG. 4, there is shown a means for carrying a plurality of electrodes through a borehole 27 filled with a conductive drilling mud for investigating adjoining earth formations 28. For purposes of explaining the operation of the FIG. 4 apparatus, the formations 28 will be assumed to be homogeneous. In FIG. 4, the carrying means includes a support means 33 which is suspended in the borehole 27 on the end of an armored multiconductor cable 31 of which the lower feet or so is covered with a suitable insulating material 32. A remote potential measuring electrode N is mounted on this insulated portion of the cable 31 and is connected by a suitable electrical conductor to electrical circuitry located in the support member33.

The electrode array also includes a central currentemitting electrode A mounted on the support member 33 with a pair of symmetrical potential measuring electrodes M, and M, located equal distances d, from A on both sides thereof. The electrodes M, and M, are electrically shorted together. A second pair of potential measure electrodes M and M (also shorted together) are mounted on the support member 33 equal distances d, from A, on both sides thereof and at a greater distance therefrom than the first measure pair M,,M,'. A third set of electrodes (also shorted together) A, and A, are mounted on the support member 33 at equal distances on both sides of the center electrode A and at greater distances therefrom than the aforementioned measure electrodes. The exterior of the portion of support member 33 on which these electrodes are mounted is constructed of a nonconductive material. The upper portion 34 of the support member 33 is a fluid-tight housing which contains the electrical circuitry to which the aforementioned electrodes are connected.

Looking at the dotted line box 34a shown to the right of the borehole 27, electrical circuitry contained within the fluid-tight housing 34 is illustrated. A constant current generator 35 supplies a constant magnitude alternating current i,, between the central current-emitting electrode A and the armor of cable 31. The potential difference 2, established between the measure electrodes M,,M,' and the remote electrode N is applied across the primary winding 36 of a transformer 37 and the potential difference e established between measure electrodes M ,M and remote electrode N is applied across the primary winding 38 of a transformer 39. The secondary windings 40 and 41 of transformers 37 and 39 respectively are series connected and coupled to the input of a high gain amplifier 42.

The transformers 37 and 39 are wound in such a manner that the input voltage e, to amplifier 42 can be expressed as follows:

(I) where C is a constant. (C will be defined later.) The amplifier 42 induces an output current i, in the secondary winding of a transformer 43, which current is emitted from the central current-emitting electrode A and returned to the two end electrodes A and A, for return to the secondary winding of transformer 43.

The gain of amplifier 42 is sufficiently high that the input voltage e thereto will be substantially zero volts. That is to say, amplifier 42 will supply sufficient auxiliary current i between the central current-emitting electrode A and the end electrodes A A, to maintain the input voltage e substantially equal to zero. Thus, setting e 0, Equation (1) can be rewritten as:

Since the voltages e and e are measured with respect to the remote potential reference electrode N, which because of its remoteness from the current-emitting electrodes, is at or substantially near zero volts, 2 and (2 will represent the potentials with respect to zero volts on measure electrodes M,,M, and M ,M Thus, Equation (2) can be written as:

where V and V,, are the potentials with respect to zero volts on measure electrodes M M and M M respectively.

Now concerning the value of C and referring to FIG. 5, there is shown the homogeneous potential distribution curve, i.e., the R R, curve of FIG. 3, reproduced. The location of the electrodes of FIG. 4 are shown along the horizontal (distance) axis of FIG. 5. Since the potential decreases as an inverse function of distance from a current source in a homogeneous medium, the potential V at the measuring electrode M, when R,, R, will be:

where V is the voltage potential at the source. Likewise, the potential V,,, at the measuring electrode M will be:

VM (M 2) 0 Combining Equations (4) and (5), we have:

Thus, comparing Equations (3) and (6), it is clear that the circuit constant C (windings ratio of transformers 37 and 39) will be equal to a /d for the FIG. 4 apparatus to produce a potential distribution as if R,, were equal to R; regardless of the value of R relative to R,.

In accordance with the operation of the feedback loop of FIG. 4 just discussed, the auxiliary current i is continuously adjusted such that the potential V,,, is always equal to (d /d V in accordance with Equation (6). By so doing, a potential distribution curve will be produced by the apparatus of FIG. 4 in substantially the form shown in FIG. 5. As discussed in connection with FIGS. 1, 2 and 3 a potential distribution curve of the shape shown in FIG. 5 represents the potential distribution for a homogeneous case, (i.e., R,,, Rf)- Thus, by enforcing this potential distribution in the borehole, a substantially spherical equipotential distribution similar to the one shown in FIG. 1 will be established regardless of the conductivity of the mud column relative to the formation. This effectively has the result of making the formation and mud column appear to be one homogeneous medium and thus any adverse affect of the mud column on the investigation of the formation is substantially eliminated.

It would perhaps be beneficial to also discuss this operation in terms of current flow. In this regard, since the survey current i emitted from the central currentemitting electrode A is returned to the relatively distant cable armor 311 and the auxiliary current i is returned to the nearby electrodes A A the auxiliary current i will tend to pass primarily through the mud column thus forcing the survey current i out into the formation for the investigation thereof. It is clear that the more conductive the mud column relative to the formation, the more auxiliary current i will be generated to flow through the mud column in order to insure that the potential relationship between the measuring electrodes M and M is as set forth in Equation (6). If R is equal to R,, then there would be very little, if any, auxiliary current i,, generated since the survey current i alone would set up the required potential distribution for the same reasons as discussed in connection with FIG. I.

Since the survey current i is always at a constant value (i.e., RMS or peak-to-peak value) through the action of the constant current generator 35, the potential existing on the central current-emitting electrode A will vary in proportion to the resistivity of the media surrounding this electrode. In terms of the potential distribution curve of FIG. 5, the entire curve will move up or down in dependence on this resistivity. To obtain a measure of the formation resistivity then, it is only necessary to measure the potential difference between two points in the borehole. Thus, assuming that the difference between the potential in the vicinity of the measure pair M ,M,' and the potential of remote electrode N is measured, that portion of a homogeneous formation lying between the equipotential line 45 which passes through the measure electrode pair M M, and the equipotential line 46 which passes through the remote reference electrode N is investigated. Since the survey current return electrode (armor of cable 31) is remotely located relative to the central current-emitting electrode A and the potential distribution curve of FIG. 5 is established in the borehole, the survey current will tend to radiate from the central electrode A in a spherical fashion provided the formation itself is homogeneous. Thus, the equipotential lines 45 and 46 will be substantially spherical in shape as shown in FIG. 4. By spacing the M M, electrodes far enough from the central electrode A the borehole will be substantially eliminated from investigation since the equipotential line 45 passes through the formation.

A measurement of the resistivity of the formation lying between the equipotential lines 45 and 46 is obtained by coupling this potential difference between electrodes M M and N via a transformer 47 to a suitable high impedance input measuring amplifier 48. The output signal from measure amplifier 48 is supplied to the input of a phase-sensitive detector 49 which derives its phase reference signal from the constant current generator 35. The phase-sensitive detector 49 generates a varying DC output signal which is proportional to that portion of the measured signal which is in-phase with the survey current i and thus is proportional to formation resistivity R This resistivity signal is supplied to the surface of the earth via a conductor pair 50 (which in reality passes through the cable 31) to a recorder 51 via suitable signal processing circuits 52. The recorder 51 is driven as a function of cable movement via a shaft 53 coupled to a rotating wheel 54. The wheel 54 engages the cable 31 so as to rotate as a function of cable movement. By this means, the resistivity signal R, is recorded by recorder 51 as a function of borehole depth to produce a log of resistivity versus depth.

It is also possible to control the potential gradient at two points in the borehole to produce the potential distribution curve of FIG. 5. The potential gradient is defined as AV/AD where AV is the difference in potential between two fairly closely spaced points and AD is the spacing between these two points-Referring now to FIG. 6, there is shown apparatus for accomplishing this in accordance with the present invention. Like the FIG. 4 embodiment, the FIG. 6 embodiment includes a plurality of electrodes mountedon a support member 55. As in the FIG. 4 embodiment, a central currentemitting electrode A is surrounded on both the upper and lower sides thereof by a symmetrical electrode array comprising in the order of their spacings from the central electrode A the electrodes M M M -M A1 A1,, MM5,, Mg-Mg and Az-Ag The M3 and M4 electrodes are spaced relatively close together and the center point therebetween is a distance d from the current source A The M and M electrodes are likewise spaced relatively close together and the center point therebetween is a distance d from the current source A (correspondingly, the M -M and M .,'-M are each spaced close together and distances of a and d from A respectively.) As in the FIG. 4 embodiment, similarly designated electrodes on both sides of the central electrode A are shorted together.

The constant current generator 56 generates a constant magnitude AC survey current i which is emitted from the central current-emitting electrode A and returns to the generator 56 via the armor of cable 31. The difference in potential AV between the potentials on the first measure electrode pair M ,M and M ,M is developed across the primary winding 57 of a transformer 58 and the difference in potential .AV developed across the measure electrode pair M ,M and M ,M is supplied to the primary winding 59 of a transformer 60. The secondary windings of transformers 58 and 60 are series connected and coupled to the input of a high gain amplifier 61. The windings of transformers 58 and 60 are set such that the input voltage e, to the monitor amplifier 61 is:

4 s-a" 1 a-4 where C is a constant determined by the winding ratio of transformers 58 and 60.. The amplifier 61 develops an auxiliary current i in the secondary winding of a transformer 62 which is emitted from the central current-emitting electrode A and returned to the auxiliary current return electrodes A A and A and A A weighted current summing network 63 sums the currents i and i which are received by the electrodes A ,A, and A ,A respectively for return to the secondary winding of transformer 62. These currents i and i, are summed in a fixed ratio so that i, ni To accomplish this, the i, current passes through a resistor of value R and the 1', current passes through a resistor of value nR,

where R is substantially higher than the resistance of the remainder of the feedback loop.

As in the FIG. 4 embodiment, the gain of amplifier 61 is sufiiciently high that enough auxiliary current i, will be supplied between the central current-emitting electrode A and the return electrodes A -A A A to maintain the input voltage e to amplifier 61 at substantially zero volts. Thus setting e 0, Equation (7) can be rewritten as:

To determine the constant C, and turning to FIG. 5, the locations of the electrodes M M M and M are represented as points on the potential distribution curve. By taking the derivative of the earlier discussed potential distribution function V A (V ID), the slope of the potential distribution curve at any point spaced a distance D from the current source is:

AV/AD A (V /D As stated earlier, AV/AD is the potential gradient. Thus, Equation (9) written for the potential gradient measured by the measuring electrodes M ,M and M ,M is:

and written for the potential gradient measured by the measuring electrodes M ,M and M M is:

Combining Equations (10) and (11), we have:

(AVSBIA 3-4) 1 2 [By making AD AD these terms drop out, although, if they were unequal, it would only change the constant portion (dB/d?) of Equation 12.] Comparing Equations (8) and (12), it is clear that the circuit constant C should be equal to d /dg to establish the desired potential distribution.

Thus, it can be seen that the FIG. 6 apparatus measures the potential gradient at one location in the borehole by means of the measure electrodes M and M and the potential gradient at a second location in the borehole by means of the measuring electrodes M and M and adjusts the auxiliary current i, to maintain a constant ratio between these potential gradients. In other words, the ratio of the slopes of the FIG. 5 curve at two borehole locations is maintained constant. By making this constant ratio equal to dfld a potential distribution curve substantially as shown in FIG. 5 will be produced by the FIG. 6 apparatus and a spherical equipotential distribution will be established in the media surrounding the electrode array to thus eliminate the mud column as a disrupting factor in the measurement of the resistivity or conductivity of the formations. The electrode spacings and the value of n can be selected to produce the desired results. (Desirably, n can be selected to be approximately the same as the squared electrode spacing ratio dg /d As with the FIG. 4 apparatus, the survey current i emitted from the central current-emitting electrode A of FIG. 6 is constant and thus the entire potential distribution curve of FIG. will increase or decrease as a function of resistivity. To provide a measure of the formation resistivity, the potentials on the center taps of primary windings 57 and 59 of transformers 58 and 60 are supplied to the input of a suitable measure amplifier 64. That component of the output signal from measure amplifier 64 that is in-phase with the survey current i is detected by a phase-sensitive detector 65 which derives its phase reference signal from the generator 56. The resulting detected signal is supplied to the surface of the earth as a varying DC signal proportional to the formation resistivity R,.

Since the potentials supplied to the measure amplifier 64 are derived from the center taps of the primary windings of transformers 58 and 60, the difference between the potential existing midway between measure electrodes M and M and the potential existing midway between measure electrodes M, and M is measured. For a homogeneous formation then, the measured resistivity is representative of the resistivity of that portion of the formation which lies between the spherically shaped equipotential line 66 which passes through a point midway between the measure electrodes M and M and a point midway between the measure electrodes M and M and the spherically shaped equipotential line 67 which passes through a point midway between the measure electrodes M and M and a point midway between measure electrodes M and M,,'. This area is shown as the hatched line area in FIG. 6. It should be noted that, if desired, another measure circuit could be utilized to provide a measure of the formation lying beyond this hatched line area in FIG. 6 by measuring the potential difference between the center tap of either primary winding 57 or 59 and the remote electrode N.

In the discussion up to this point, it has been assumed that the formation itself is relatively homogeneous, the only complicating factor being a highly conductive mud column. The apparatus of FIGS. 4 and 6 each operates to substantially minimize the adverse effects of this conductive mud column on the formation resistivity measurements. However, as is many times the case, the formations are relatively nonhomogeneous. This non-homogeneity can take the form, for example, of boundaries between the formation beds where one formation has a resistivity substantially different from an adjoining formation. Additionally, in permeable formations, the drilling mud will invade the formation thus displacing all or a portion of the naturally occurring formation fluids contained in the invaded portion. In many cases, the resistivity of the invaded formation zone will be different from the resistivity of the noninvaded zone and sometimes the contrast will be quite large.

Referring to FIG. 7, there is shown a diagram of these various zones. A logging tool 70 is shown in a borehole 71 immersed in a mud column with a resistivity R,,,. The invaded zone having a resistivity R is the zone next adjacent to the borehole 71 and the noninvaded zone having a resistivity R, is that portion of the formation furthest removed from the borehole 71. The radius of invasion r, is the distance between the center of the borehole and the interface between the invaded and noninvaded zones. Inaddition, nonhomogeneity occurs when formation beds having a resistivity, R, difierent from the resistivity of the zone presently under investigation are located adjacent the investigated zone.

As discussed earlier, the apparatus illustrated in FIGS. 4 and 6 will set up a potential distribution curve illustrated in FIG. 5 along the borehole axis as well as radially into the formation. However, if the formation itself is not homogeneous, an exploring device attempting to establish the potential distribution for a homogeneous formation will, in effect, be attempting to establish an unnatural potential distribution which may cause measurement errors, especially where the radius of invasion r, is relatively shallow and the resistivity contrast relatively high.

Thus referring to FIG. 8, there is shown another form of the present invention wherein the resistivity can be measured for substantially all formation conditions and yet the effect of a highly conductive mud column on such measurements will be substantially minimized. This is accomplished by maintaining a constant potential distribution in only one region on each side of the central current-emitting electrode. In this FIG. 8 embodiment, a support member is supported in the borehole on the end of the cable 71. The support member 70 includes an electrode array comprising a central current-emitting electrode A and a plurality of symmetrically disposed electrodes on the upper and lower sides of the central A electrode. These symmetrical electrodes, in the order of their spacings from the central electrode A are M -M A -A M -M, and M -M As in the prior embodiments, those electrodes having the similar designation (prime or no prime) are shorted together.

In FIG. 8, there is also shown a coil system or induction logging system mounted on the support member 70. Only two illustrative coils 73 and 74 are shown, which coils are embedded in a layer of rubber material (not shown) which encircle a center mandrel 75. The center mandrel 75 and coils are enclosed by an outer sleeve member 76 of cylindrical shape. (Part of this sleeve member 76 is broken away in FIG. 8 for purposes of showing the coils located therebelow.)

Concerning the electrode construction, the A A and A, current electrodes individually comprise a series of small rectangular metal plates which are positioned to encircle the circumference of the sleeve member 76. The rectangular plates of each electrode are connected together by means of a closed loop of slightly resistive wire which encircles the sleeve member 76 immediately below the rectangular plates. The measuring electrodes M M M M M and M on the other hand, are constructed in a similar manner except that small metal discs or buttons are used in place of the rectangular plates since these measuring electrodes do not emit or receive a large amount of current as do the current electrodes A A, and A,'. This electrode construction could, of course, be used with any electrode system discussed herein and is not limited to just the FIG. 8 embodiment.

By utilizing the electrode construction shown in FIG. 8, the electrodes of the electrode array will not take the form of a closed conductive loop and thus will have a negligible influence on the operation of the induction logging system. In this manner, an electrode array and a coil array can both be utilized simultaneously on the same support member. For a more detailed explanation of the construction of the coils and electrodes of FIG.

8, see US. Pat. No. 3,329,889 granted to D. R. Tanguy on July 4, 1967.

Now concerning the electronic circuitry contained within the fluid-tight.housing 70, refer to the circuitry shown within the dotted line box 70a, which corresponds to the fluid-tight housing 70. An oscillator 78 couples a constant reference voltage V across the secondary winding 79 of a transformer 80. One side of the second winding 79 is connected to the center tap of the primary winding 81 of a transformer 82. The measure electrodes M ,M are connected to one side of the primary winding 81 and measure electrodes M ,M are connected to the other side thereof. The other side of secondary winding 79 is connected to the measure electrode M and M via the primary winding 83 of a transformer 84. The secondary winding of transformer 84 is coupled to the input of a high gain amplifier 85 which supplies sufficient auxiliary current i between the control current-emitting electrode A and the current return electrodes A,,A by way of a transformer 86 such as to maintain the potential e across the primary winding 83 of transformer 84 at substantially zero volts.

The equation for the voltage e developed across the primary winding 83 can be expressed as:

where V is the voltage on the measuring electrodes M and M and V is the voltage existing on the center tap of primary winding 81 of transformer 82, and thus the potential existing at a point between the measuring electrodes M, and M (and likewise M and M Since the action of amplifier 85 supplying auxiliary current i,, between the electrodes A and A -A maintains the voltage e developed across primary winding 83 at substantially zero volts, Equation (13) can be rewritten as:

u VM ref Equation (14) states that this feedback circuit just described will operate to maintain the potential difference between the measuring electrodes M ,M and a point intermediate the measuring electrodes M,,M,' and M ,M at a constant reference level. This reference level is the constant reference voltage V developed across the secondary winding 79 of transformer 80.

A signal proportional to the potential difference between the measure electrodes M ,M and M ,M is supplied to an amplifier 87 by way of the transformer 82 and the zero gradient contact of a double-throw switch 180. Amplifier 87 supplies sufficient survey current i between the central current-emitting electrode A and the outer metal sheath of the fluid-tight housing 70 via a transformer 88 to maintain a zero potential difference between the measure electrodes M ,M and MmMz To summarize the operation of that portion of the FIG. 8 apparatus described thus far, refer to FIGS. 8' and 9 in conjunction where FIG. 9 shows the current flow pattern produced by the apparatus of FIG. 8 in a typical situation. The amplifier 85 produces sufficient auxiliary'current i,I to maintain a constant potential difference between the equipotential line 94 which passes through the measuring electrodes M and M andthe equipotential line which passes through a point between the measuring electrodes M M, and M,,M,'.

The amplifier 87, on the other hand, supplies sufficient survey current i between the central currentemitting electrode A and the outer metallic sheath of the housing 77 to establish a substantially zero potential gradient (or zero electric field) between the measuring electrodes M M, and M,,M,'. By so doing, it is insured that no current will flow through the borehole in a direction parallel to the borehole axis in the vicinity of the measuring electrodes M M and M ,M By this means, the borehole will be electrically plugged at the measuring electrodes M M, and M ,M the same as if electrical insulators were placed in the borehole at these two locations. This then prevents the survey current i from shorting through the borehole to produce the effect discussed in connection with FIG. 2.

Since the auxiliary current return electrodes are inside the electrical plugs (points of zero gradient) on both sides of the central current-emitting electrode A and the survey current return electrode (metal sheath 77) is outside the electrical plug, the auxilary current i will flow primarily in the borehole thus forcing the survey current i out of the borehole and into the formation, as shown in FIG. 9. The auxiliary current i,, and survey current i will tend to break in opposite directions in the vicinity of the zero gradient locations with the auxiliary current i being pulled back to the auxiliary return electrodes A and A and the survey current i tending to flow in the opposite direction away from the current electrodes A A, and A,'.

It should be noted here that even though the auxiliary current i is no longer forcing the survey current i away from the conductive fluid-filled borehole at borehole locations above the measuring electrodes M,-M and below the measuring electrodes M -M the dimensions of the borehole become important as the survey current moves further and further away from the central current-emitting electrode A Thus, even though the borehole may be far more conductive than the formation, as the survey current moves further away from the A electrode, it will tend to spread out into the formation. By so doing, the volume of formation through which the survey current i passes becomes substantially greater than the volume of the mud column as the distance from the central electrode A increases and thus the overall resistance offered to the survey current by the formation becomes less than the overall resistance of the mud column. Thus, by appropriately placing the measuring electrodes M M, and M ,M the survey current will not have a tendency to return to the borehole at a distance away from the A electrode where the effect of the auxiliary current has diminished.

The apparatus of FIG. 8 can produce a relatively shallow investigation of the formation. How shallow depends on the spacings between electrodes and the dimensions of the electrodes. This can be seen in FIG. 9 by observing that the auxiliary current i,, is constrained primarily to the mud column and thus the survey current i,, does not have a tendency to penetrate deeply into the formation. This is contrasted with the so-called focused electrode systems which force survey current deeply into a formation through the action of focusing current emitted from nearby focusing electrodes, which focusing current is in-phase with the sur vey current. Electrode systems of this type can be found in U. S. Pat. No. 2,712,627 granted to H. G. Doll on July 5, 1955 or U. S. Pat. No. 3,031,612 granted to M. F. Easterling on Apr. 24, 1962. It should be noted here that the potential on the auxiliary electrodes A and A of the apparatus of the present invention is 180 out-of-phase with the potential on the central current-emitting electrode A as compared with conventional focused electrode systems where the potentials of the survey and focusing electrodes are in-phase.

The radial depth of investigation of the apparatus of the present invention can be selected by properly selecting the placement and size of the electrodes. Thus, for example, by moving the measuring electrodes M M and M ,M of FIG. 8 closer to the central electrode A the radial depth of investigation becomes less. However, if moved too close, the borehole effect becomes a problem (i.e., much of the survey current would return through the borehole if it is conductive enough). Obviously, a compromise between the two conflicting considerations is in order if shallow investigation is desired.

Additionally, the placement of the survey current return electrode is somewhat of a factor in determining the radial depth of investigation if this electrode is relatively close to A As shown in FIG. 8, the current re turn for the survey current i is the metal housing 77 which is located relatively close to the current-emitting electrode A but exterior of the measuring electrodes M M, and M ,M By so doing, the survey current i is prevented from penetrating too deeply into the formations. Recalling that equipotential lines are perpendicular to current flow lines, the equipotential lines of FIG. 8 will no longer be spherically shaped since the survey current i is being drawn toward the housing 77. Thus, the depth of investigation will tend to be relatively shallow as shown by the equipotential lines 94 and 95 in FIG. 8. If desired however, this current return could be placed on the armor of the cable 71 as in the FIGS. 5 and 6 embodiments or these embodiments could have the i current return electrode positioned as in the FIG. 8 embodiment.

, After extensive research and experimentation, a desirable spacing of the electrodes has been found to be: A to M and M and 4.4 inches; A, to A, and A and 8 inches; A to M, and M and 19 inches; A to M and M and 23 inches; and A to the housing 77 60 inches with the housing 77 extending upward another 2 feet or so.

Since the equipotential lines 94 and 95 of FIG. 8 have a constant potential difference therebetween, it is clear that to obtain a measure of this hatched line portion of the formation shown in FIG. 8, it is merely necessary to measure the survey current which passes therethrough to obtain a measure of the conductivity thereof. To accomplish this, a very low resistance measure resistor 89 is inserted in series with the secondary winding 90 of transformer 88 so that a voltage proportional to the magnitude of survey current i is developed across this resistor 89. This measure voltage is applied to a high input impedance measure amplifier 91 whose output signal is supplied to a phase-sensitive detector 92. The phase-sensitive detector 92 detects that portion of the measured voltage from measure amplifier 91 which is in-phase with the oscillator 78 output signal and provides a varying DC signal proportional to the conductivity a of the invaded zone (since the investigation is very shallow) to the surface of the earth.

It should be noted at this point that even though the survey current i was generated in response to the potential difference between the measure electrodes M,,M and M ,M and the auxiliary current i, was generated in response to the potential difference between the M M measuring electrodes and a point between the M M and M ,M electrodes, these two functions could be reversed. If this change were made, it would be necessary to switch the return electrodes which are connected to the secondary windings of transformers 86 and 88 so that the survey current i will be returning from the housing 77 to the secondary winding of transformer 86 and the auxiliary current returning from the electrodes A A, to the secondary winding of transformer 88. Measure resistor 89 would then be inserted in series with the secondary winding of transformer 86 to obtain a measure of the survey current. Additionally, even though a constant voltage system has been shown in FIG. 8 (which could be adapted to the FIGS. 4 and 6 embodiments as well), a constant current system could just as easily be utilized.

It would be desirable at this point to briefly compare the form of the invention depicted in FIG. 8 with that depicted in FIGS. 4 and 6. The form of the invention illustrated in FIGS. 4 and 6 utilizes two points of control on each side of the central A electrode for maintaining a potential distribution substantially as shown in FIG. 5. Thus, the FIG. 4 apparatus maintains a constant ratio of the potentials measured by electrodes M M, and M ,M while the apparatus of FIG. 6 maintains a constant ratio of the potential gradients at the electrode pairs M ,M M ,M and M ,M,,, M ,M to achieve the potential distribution curve of FIG. 5 The form of the invention shown in FIG. 8, on the other hand, only utilizes one point of control on each side of the central A electrode, namely, a zero potential gradient at the measuring electrodes M M, and M,,M By so doing, the borehole is electrically plugged thus forcing the survey current to flow outwardly into the formation and not short through the borehole.

Thus, in a homogeneous formation, a potential distribution curve somewhat similar to that shown in FIG. 5 will be produced by the FIG. 8 apparatus in much the same manner as in the FIGS. 4 and 6 apparatus. However, when the formation is relatively nonhomogeneous, the FIGS. 4 and 6 apparatus will attempt to establish the homogeneous case or type potential distribution in this nonhomogeneous formation which may tend to produce errors in the measurements while the FIG. 8 apparatus will allow the potential distribution in the formation to take its natural course, but without undue influence from the mud column.

It will be recalled from a discussion of FIG. 5 that to produce a spherical equipotential distribution in a homogeneous formation, the potential distribution depicted in FIG. 5 must be established in the formation. However, since the system of FIG. 8 establishes a region of zero potential gradient on both sides of the central -A electrode, it can be seen that even if R R, and the formation itself is homogeneous, the FIG. 8 apparatus will not establish a spherical equipotential distribution. However, if the monitoring electrodes M ,M '-M ,M are located at a sufi'icient distance from the central survey electrode A the potential distribution will be so near spherical that the simplicity of the electronic circuitry is worth the slight error. (Note: This error can be corrected for by suitable correction charts.) The reason for this simplicity is that it is easier to maintain a zero potential gradient than a finite value thereof.

If however, an exact spherical distribution were considered desirable for the homogeneous formation case, the FIG. 8 circuitry could be modified to bring this about by establishing a finite potential gradient at the monitoring electrodes M,,M '-M ,M This could be accomplished, for example, by adding a constant reference amplitude signal derived from the oscillator 78 by way of another secondary winding 79a of the transformer 80 with the voltage developed across the secondary winding 82a of transformer 82. Thus, in FIG. 8, the switch 180 would be positioned in the finite gradient position so that the i amplifier 87 will cause sufficient survey current to be emitted from the A electrode to maintain V equal to the reference voltage developed across the secondary winding 79a. The value of this gradient can be set by suitably selecting the number of turns for secondary winding 79a. Desirably, this gradient should be maintained at a value which will maintain a spherical equipotential distribution in a homogeneous formation. Thus, referring to FIG. 5, if the measuring electrodes M M and M ,M are at the locations designated 184 and 185, the reference voltage developed across the secondary winding 79a would be set at a value to produce a gradient defined by the slope of the curve at the region defined by the points 184 and 185.

In this alternative FIG. 8 case, the current flow through the borehole would be regulated to a desired degree thus preventing very much of the survey current from shorting through a conductive mud column. Thus, in a homogeneous formation, the amount of current allowed to pass through the borehole would be no greater than the current flow in any other direction thus establishing the spherical equipotential situation of FIG. 1. However, if the formation is nonhomogeneous, the potential distribution in-the formation can take any form since the potential distribution for a homogeneous formation will not be enforced along the length of the electrode array. Effectively, then, in this modified version of FIG. 8, the borehole is electrically plugged at the measuring measuring electrodes M M, and M ,M except for a small flow of survey current.

It should be noted at this point that even though the FIG. 8 apparatus maintains a constant potential difference between two points along the electrode array and the FIGS. 4 and 6 apparatus maintain a constant survey current flow, these controls cannot be classed as controls necessary for performing the objects of the present invention. Instead, these constant voltage and constant current controls are merely controls of convenience rather than necessity. That is to say, by maintaining either the voltage or current constant, it is only necessary to measure the other variable electrical parameter to obtain a measure of either the resistivity or conductivity of the formation. The FIGS. 4, 6 and 8 apparatus would work just as well if neither the voltage nor current were maintained constant and the variations in both parameters measured so that a ratio of voltage to current would produce a resistivity measurement. Moreover, in the FIGS. 4, 6 and 8 embodiments,

either the i, or i current could be utilized to control the potential distribution.

It should also be noted that the equipotential plots discussed earlier, as well as those to be discussed later, are estimates of the real situation since it is impossible to measure them point by point in the formation. Also, these equipotential plots are drawn for homogeneous formations and will tend to become somewhat distorted in nonhomogeneous formations.

As set forth above, it is only necessary to have one region of control on each side of the central A electrode and one control parameter to produce the desired results in accordance with the present invention. However, if desired, there could be a plurality of zero (or finite) potential gradient control regions to achieve the objects of the present invention. Thus, referring to FIG. 10, there is shown a more elaborate embodiment of the present invention utilizing the technique set forth in FIG. 8. As in the prior embodiments, the central current-emitting electrode A is surrounded on both the upper and lower sides by a symmetrical electrode array comprising in the order of their distance from the central electride A the electrodes ol o; A1,A1'; b l M2,M2I; A2,A2'; Ss 3 M ,M and M ,M The M M, and M ,M measuring electrodes are located relatively close together as are the measuring electrodes M ,M and M ,M As before, the upper portion of the support member 102 comprises a fluid-tight housing 103 containing the electrical circuitry.

This electrical circuitry is shown in the dotted line box 103a in FIG. 10 and includes a constant current generator 104 which supplies a constant value of survey current i between the central current-emitting electrode A and the armor of the cable 31. A high gain amplifier 105 is responsive to the potential gradient between the measuring electrodes M,,M, and M ,M via a transformer 106 for supplying sufficient auxiliary current i between the central current-emitting electrode A and the current return electrode A A, via a transformer 107. As before, the magnitude of this auxiliary current i is sufficient to maintain the potential gradient between the measuring electrodes M M and M ,M at substantially zero volts. This operation is the same as in the FIG. 8 embodiment. However, in this FIG. 10 embodiment, another high gain amplifier 108 is responsive to the potential gradient between the measuring electrodes M ,M and M ,M via a transformer 109 to .supply sufficient auxiliary current i between the current electrodes A ,A and A ,A so as to maintain the potential gradient between the measuring electrodes M ,M and M ,M at substantially zero volts. Thus, it can be seen that the apparatus of FIG. 10 operates to plug the borehole at two separate points on each side of the central current-emitting electrode A By this means, the survey current i emitted from the central electrode A will be forced out of the borehole into the formation for a greater interval than would be possible with the FIG. 8 apparatus. This could be advantageous for example if the mud column were so much more conductive than the formation that a great amount of the survey current i would tend to be drawn into the mud column at a point just beyond the first measuring electrodes M ,M, and M ,M Additionally, it has been found that the radial depth of investigation will be slightly decreased if this second loop is utilized. 

1. AppAratus for measuring a characteristic of earth formations traversed by a borehole, comprising: a plurality of electrodes including at least one currentemitting electrode, a pair of measuring electrodes located on each side of said at least one current-emitting electrode, first current electrodes including a pair of electrodes with one each located between each of said measuring electrode pairs and said at least one current-emitting electrode, and a second current return electrode spaced apart from said first current return electrodes; means for mounting said electrodes for passage through a borehole; means responsive to the potential difference between the potentials of said measuring electrode pairs for supplying a first current between said at least one current-emitting electrode and one of said first or second current return electrodes of sufficient magnitude to maintain said potential difference at a substantially constant value; means responsive to a second potential difference between a potential at a first point lying between said first current return electrodes and a potential at a second point between said measuring electrodes for supplying a second current between said at least one current-emitting electrode and the other of said first or second current return electrodes of sufficient magnitude to maintain said second potential difference at a substantially constant level; and means responsive to one of said first or second currents for obtaining a measure of a formation characteristic.
 2. Apparatus for measuring a characteristic of earth formations traversed by a borehole, comprising: a plurality of electrodes including at least one current-emitting electrode, a pair of measuring electrodes located on each side of said at least one current-emitting electrodes, auxiliary current return electrodes located between each of said measuring electrode pairs and said at least one current-emitting electrode, auxiliary measuring electrodes with one each located between said at least one current-emitting electrode and each of said auxiliary current return electrodes, and a survey current return electrode spaced apart from all of said other electrodes; means for mounting said electrodes for passage through a borehole; means responsive to the potential difference between the potentials of said measuring electrode pairs for supplying a survey current between said at least one current-emitting electrode and said survey current return electrode of sufficient magnitude to maintain said potential difference at a substantially fixed value; means responsive to a second potential difference between the potential of said auxiliary measuring electrodes and a potential of a point lying between said measuring electrodes for supplying an auxiliary current between said at least one current-emitting electrode and said auxiliary current return electrodes of sufficient magnitude to maintain said second potential difference at a substantially constant level; and means responsive to said survey current for obtaining a measure of the conductivity of a formation.
 3. A method of measuring a characteristic of earth formations traversed by a borehole, comprising: carrying a plurality of electrodes through a borehole, said electrodes including at least one current-emitting electrode, a pair of measuring electrodes located on each side of said at least one current-emitting electrode, first current return electrodes including a pair of electrodes with one each located between each of said measuring electrode pairs and said at least one current-emitting electrode, and a second current return electrode spaced apart from said first current return electrodes; measuring the potential difference between the potentials of said measuring electrode pairs; supplying a first current between said at least one current-emitting electrode and one of said first or second current return electrodes in response to said measured potential difference to maintaiN said potential difference at a substantially constant value; measuring a second potential difference between a potential at a point lying between said first current return electrodes and a potential of a point lying between said measuring electrodes; supplying a second current between said at least one current-emitting electrode and the other of said first or second current return electrodes in response to said second measured potential difference to maintain said second potential difference at a substantially constant level; and measuring one of said first or second currents to obtain an indication of a formation characteristic. 